for all x, y [member of] D(T) and n [greater than or equal to]
[n.sub.0].

Remark 1.3. If we set n = [n.sub.0] = 1 in (1.1), we get the
definition of a strongly pseudocontractive map. An example of a map
which is not strongly pseudocontractive but which is strongly
successively pseudocontractive can be found in [1].

One of the effective methods for approximating fixed points of an
operator T : D(T)[subset] X [right arrow] X is the Mann iteration process [10], starting with arbitrary [x.sub.0] [member of] D(T) and for
n [greater than or equal to] 0 defined by

where {[a.sub.n]}, {[b.sub.n]}, {[c.sub.n]} are sequences in [0,1]
such that [a.sub.n] + [b.sub.n] + [c.sub.n] = 1 and {[u.sub.n]} is a
bounded sequence in D(T). Clearly, this iteration process with errors
has (1.3) as its special case. Furthermore, this Mann iterative process
is the same as the one introduced by Liu [12] if [c.sub.n] = 1 for all
n. However, due to the defects of the one introduced by Liu [12] as
pointed out in [], we are interested in Xu's more general Mann
iterative process of (1.4).

Now, we consider the following iteration introduced by Schu [9] for
approximation of fixed points of Lipschitz pseudocontractive maps.

This iteration is known as modified Mann iteration with errors,
where {un}e X is a bounded sequence and are error terms and {[a.sub.n]},
{[b.sub.n]}, {[c.sub.n]} are real sequences in [0,1] satisfying some
conditions. Replacing Tn by T in (1.5) one obtains Mann iteration with
errors in (1.4).

Liu [12] proved that the Mann iteration process defined by (1.3)
converges strongly to the unique fixed point of a Lipschitzian strictly
pseudocontractive mapping. Sastry and Babu [5] proved that any fixed
point of a Lipschitzian pseudocontractive self mapping of a non-empty
closed convex subset K of a Banach space X is unique and may be norm
approximated by Mann iterative procedure (1.3). Their result generalized the result of Liu [12] in the sense that, the assumption that K is
bounded was removed and a general convergence rate estimate was
provided. Recently, Rafiq [7] extended the above results to the Mann
iteration sequence with errors (1.4) in a real Banach space.

In this paper, our purpose is to show that the more general
modified Mann iteration sequence with errors converges to the unique
fixed point of T if T : X [right arrow] X is a uniformly continuous
strongly successively pseudocontractive mapping with a bounded range or
T : X [right arrow] X is uniformly Lipschitzian and strongly
successively pseudocontractive mapping without necessarily having a
bounded range. Our results extend and improve the results of Rafiq [7],
Liu [12], Sastry and Babu [5] Mogbademu et al. [11]. Furthermore, we are
able to obtain a more general and better convergence rate than those
obtained by Liu [12], Sastry and Babu [5], mogbademu et al [11] and the
very recent estimate of Ciric et al. [3].

Theorem 2.1. Let X be a real Banach space and T : X [right arrow] X
be a uniformly continuous and strongly pseudocontractive mapping with a
bounded range. Let p be a fixed point of T and the modified Mann
iteration with errors {[x.sub.n]} be defined by (1.5) with {[a.sub.n]},
{[b.sub.n]}, {[c.sub.n]} [subset] [0, 1] satisfying the following
conditions:

(i)[[infinity].summation over (n = 0)] [b.sub.n] = [infinity]

(ii) [c.sub.n] = 0

(iii) [lim.sub.n[right arrow][infinity]] [b.sub.n] = 0

where {[u.sub.n]} is a bounded sequence in X. Then the sequence
[{[x.sub.n]}.sup.[infinity].sub.n = 0] converges strongly to the unique
fixed point of T.

Proof. Since p is a fixed point of T, then the set of fixed points
F(T) of T is nonempty. Set

It is clear that [parallel][x.sub.0] - p[parallel][less than or
equal to] M. Suppose that [parallel][x.sub.n] - p[parallel][less than or
equal to] M, a similar argument leads to [parallel][x.sub.n+1] -
p[parallel][less than or equal to] M, since

Corollary 2.2. [7] Let T : K [right arrow] K be a uniformly
continuous and strongly pseudocontractive mapping with a bounded range.
Let p be a fixed point of T and let the Mann iterative scheme
[{[x.sub.n]}.sup.[infinity].sub.n=0] be defined by

and {[u.sub.n]} is a bounded sequence in X. Then the sequence
[{[x.sub.n]}.sup.[infinity].sub.n=0] converges strongly to the unique
fixed point p of T

Theorem 2.3. Let T : X [right arrow] X be a strongly successively
pseudocontractive (without having a bounded range) and uniformly
Lipschitzian with L [less than or equal to] 1. Let p be a fixed point of
T and {[x.sub.n]} be defined by (1.5a) with {[b.sub.n]} [subset] [0, 1],
satisfying the conditions:

In order to have a more detailed analysis of the convergence rate
estimate of our A value and the ones obtained by Liu [12], Sastry and
Babu [5], Mogbademu et al [11] and Ciric et al [3], we designed a
[C.sub.++] program whose input are the specific parameters (k, [eta] and
L) and which produces as output a number of iterates, depending on the
stopping criterion adopted. The significant results are the following:
From Theorem 2.3 of this paper: